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# Useful fact varx e x 2 e 2 x let x be a geometric

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Unformatted text preview: omness of outcomes. Useful fact: Var(X ) = E X 2 − E 2 [X ] Let X be a Geometric random variable. pX (k ) = P (X = k ) = (1 − p )k −1 p , if k = 1, 2, · · · ; 0, otherwise; 1 Var(X ) = p12 − p (Hint: let S = E X 2 . Compute S − (1 − p )S ; you may ﬁnd E [X ] handy in simplifying the calculation.) M. Chen ([email protected]) ENGG2430C lecture 6 4 / 17 Bernoulli Random Variable Experiment: Flip a coin and obtain two possible outcomes: HEAD or TAIL Deﬁne a Bernoulli random variable X as X (ω ) = 1, ω = outcome is ’HEAD’ 0, ω = outcome is ’TAIL’ The PMF is pX (x ) = p, if x = 1 1 − p , otherwise The expectation and variance of a Bernoulli r.v. X : E (X ) = 1 · p + 0 · (1 − p ) = p Var (X ) = E X 2 − E 2 (X ) = p (1 − p ) M. Chen ([email protected]) ENGG2430C lecture 6 5 / 17 Properties of Expectation * Expectation is a linear operation on the values x ’s. E [X ] = ∑ xpX (x ). (A weighted sum of x ’s.) x Let α and β be constant: E [α ] = ? α E [α X ] =? α E [X ] E [α X + β ] =? α E [X ] + β Let Y = g (X ) where g (·) is a function, compute E [Y ]. E [Y ] = ∑ g (x )pX (x ) x M. Chen ([email protected]) ENGG2430C lecture 6 6 / 17 Properties of Variance* Variance is a...
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